On Aristotelian Logic

In Περὶ Ἑρμηνείας Aristoteles defines the basic terms of his logic to be single predicates, each of single subjects see for example the Stanford Encyclopedia of Philosophy

This excludes all predicates of higher arity, e.g. two-place relations. What effects does this restriction imply for the statements that can be logically analysed and discussed?

Timelessness

As every predicate has only one argument, then if it holds at all, it must hold eternally, because there is no possible second parameter to serve as a timestamp or interval of validity.

So statements like Socrates is alive, coffee is ready, the emperor of france is bald are either true or false; and if true at the time when their validity is checked, then they must be true eternally.

The discourses using this logic are restricted to timeless expressions. It is impossible to examine processes.

Unrelatedness and separation

Since only unary relations are allowed, statements such as Socrates is married to Xanthippe are inexpressible. Aristotle is therefore unable to infer the valid Xanthippe is married to Socrates, even if he would somehow know the general rule If A is married to B, then B is married to A, (or “Being married to” is a symmetric relation, which is expressible in his logic).

Subjects can only be treated in isolation. In the resulting ontology they are cut off from each other and from all context by the restrictions of the underlying logic.

Inexpressability of basic mathematical concepts
Euclid's Geometry

Geometry concerns intersections of two (or more) lines, angles formed by two lines, figures enclosed by three or more lines, etc. For example Euclid's fifth postulate begins with if two straight lines lying in the same plane intersect a third [...]. The relation “line A intersects line B” cannot be formed from one-place predicates. Although Aristotle uses these in his examples in Prior Analytics, Posterior Analytics and Meteorologica, he'd be unable to formulate them in his Logic.

Functions

The fundamental concept of a function cannot be expressed in Aristotelian logic, because it requires at least two-place relations (“f at x has value f(x)”). So even eternal laws of Physics — like Newtons law of gravity — can (ironically) not be stated with the logic of the author of Φυσικὴ ἀκρόασις.

Equivalence Relations

Since the late 19th century, many mathematical structures are formally defined by equivalence relations on sets or classes see for example The Search for Mathematical Roots, 1870-1940 by Ivor Owen Grattan-Guinness, Princeton Univ. Press ,chapter 5, Paragraph 5.3.6.

These are two-place relations which are

1. reflexive : for every a: R(a,a)
2. symmetric : R(a,b) ⇒ R(b,a)
3. transitive : R(a,b) and R(b,c) ⇒ R(a,c)

For example the elements of the cyclic group of numbers modulo thirteen are defined as the sets of all numbers having the same residue after division by 13. So the thirteen elements are the quotient sets Z/R

• n : R(n,0) : (..., -13, 0, 13, 26, 39, ...)
• n : R(n,1) : (..., -12, 1, 14, 27, 40, ...)
• n : R(n,2) : (..., -11, 2, 15, 28, 41, ...)
• ..
• n : R(n,11) : (..., -2, 11, 24, 37, 50, ...)
• n : R(n,12) : (..., -1, 12, 25, 38, 51, ...)

as defined by this relation.

Since two-place relations cannot be used, most of modern mathematics is outside the possible subjects of Aristotelian logic, although practically all mathematical statements are timeless.

In the light of these findings, the broad adoption of Aristoteles' logic in Western European theology and philosophy could be judged an impediment to reasoning about practically any non-trivial subject.

This may sound harsh, but it was remarked before by much more distinguished writers:

The doctrine of the individual independence of real facts is derived from the notion that the subject-predicate form of statement conveys a truth which is metaphysically ultimate. According to this view, an individual substance with its predicates constitutes the ultimate type of actuality. If there be one individual, the philosophy is monistic; if there be many individuals, the philosophy is pluralistic. With this metaphysical presupposition, the relations between individual substances constitute metaphysical nuisances: there is no place for them. Accordingly — in defiance of the most obvious deliverance of our intuitive 'prejudices' — every respectable philosophy of the subject-predicate type is monistic. The exclusive dominance of the substance-quality metaphysics was enormously promoted by the logical bias of the mediaeval period. It was retarded by the study of Plato and of Aristotle.
A.N.Whitehead, Process and Reality(1929), p. 137

Or:

Such investigations show very soon that traditional Aristotelian scholastic logic is quite inadequate for this purpose [of finding a constitutive theory].
Neurath, Carnap and Hahn in Wissenschaftliche Weltauffassung. Der Wiener Kreis (2012), F. Stadler.

Tue, 17 May 2022